Template:Grp model: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
(Redirected page to Recurrent Event Data Analysis)
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
=== The GRP Model  ===
#REDIRECT [[Recurrent Event Data Analysis]]
 
In this model, the concept of virtual age is introduced. Let&nbsp;<math>{{t}_{1}},{{t}_{2}},\cdots ,{{t}_{n}}</math> represent the&nbsp;successive failure times and let <math>{{x}_{1}},{{x}_{2}},\cdots ,{{x}_{n}}</math> represent the time between failures ( <math>{{t}_{i}}=\sum_{j=1}^{i}{{x}_{j}})</math> . Assume that after each event, actions are taken to improve the system performance. Let <span class="texhtml">''q''</span> be the action effectiveness factor. There are two GRP models:
 
Type I:
 
::<span class="texhtml">''v''<sub>''i''</sub> = ''v''<sub>''i'' − 1</sub> + ''q''''x'''''<b><sub>''i''</sub> = ''q'''</b>''t''<sub>''i''</sub></span>
 
Type II:
 
::<math>{{v}_{i}}=q({{v}_{i-1}}+{{x}_{i}})={{q}^{i}}{{x}_{1}}+{{q}^{i-1}}{{x}_{2}}+\cdots +{{x}_{i}}</math>
 
where <span class="texhtml">''v''<sub>''i''</sub></span> is the virtual age of the system right after <span class="texhtml">''i''</span> th repair. The Type I model assumes that the <span class="texhtml">''i''</span> th repair cannot remove the damage incurred before the ith failure. It can only reduce the additional age <span class="texhtml">''x''<sub>''i''</sub></span> to <span class="texhtml">''q''''x'''''<b><sub>''i''</sub></b></span> . The Type II model assumes that at the <span class="texhtml">''i''</span> th repair, the virtual age has been accumulated to <span class="texhtml">''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub></span> . The <span class="texhtml">''i''</span> th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to <span class="texhtml">''q''(''v''<sub>''i'' − 1</sub> + ''x''<sub>''i''</sub>)</span> .
 
The power law function is used to model the rate of recurrence, which is:
 
::<span class="texhtml">λ(''t'') = λβ''t''<sup>β − 1</sup></span>
 
The conditional <span class="texhtml">''p''''d''''f''</span> is:
 
::<math>f({{t}_{i}}|{{t}_{i-1}})=\lambda \beta {{({{x}_{i}}+{{v}_{i-1}})}^{\beta -1}}{{e}^{-\lambda \left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i-1}^{\beta } \right]}}</math>
 
MLE method is used to estimate the model parameters. The log likelihood function is [[[Appendix: Weibull References|28]]]:
 
::<math>\begin{align}
  & \ln (L)= n(\ln \lambda +\ln \beta )-\lambda \left[ {{\left( T-{{t}_{n}}+{{v}_{n}} \right)}^{\beta }}-v_{n}^{\beta } \right] \\
  & -\lambda \underset{i=1}{\overset{n}{\mathop \sum }}\,\left[ {{\left( {{x}_{i}}+{{v}_{i-1}} \right)}^{\beta }}-v_{i}^{\beta } \right]+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{x}_{i}}+{{v}_{i-1}}) 
\end{align}</math>
 
where <span class="texhtml">''n''</span> is the total number of events during the entire observation period. <span class="texhtml">''T''</span> is the stop time of the observation. <span class="texhtml">''T'' = ''t''<sub>''n''</sub></span> if the observation stops right after the last event.

Latest revision as of 07:49, 29 June 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Grp_model&oldid=25812"